Analysis of variance (ANOVA) is a method for testing the hypothesis that there is no difference between two or more population means.

2.  Introduction
 What is ANOVA
 Basic principle of ANOVA
 Assumptions in ANOVA
 ANOVA techniques
› One-way (or single factor) ANOVA
› Two-way ANOVA
 Types of two way ANOVA
 Graphical method for interactions in two-way design
 Analysis of Co-Variance(ANCOVA)
› Why ANCOVA
 ANCOVA techniques
 Assumptions in ANCOVA
 References

3.  Analysis of variance (ANOVA) is a method for
testing the hypothesis that there is no difference
between two or more population means.
 The ANOVA technique enables us to perform the
simultaneous test and as such is considered to be an
important tool of analysis in the hands of a
researcher.
 The significance of the difference of means of the
two samples can be judged through either z-test or t-
test.

4.  z- test is applied to find out the degree of
reliability of a statistics in case of large sample.
 z- test is based on the normal probability
distribution, and is used for judging the
significance of several statistical measures,
particularly mean.

5.  t-test is used to test the null hypothesis that the
population means of two groups are the same.
 t-test with two samples is commonly used with
small sample sizes, testing the difference
between the samples when the variances of two
normal distributions are not known.
 t-test is also be used for judging the significance
of the coefficients of simple and partial
correlations.

6.  When there are more than two means, it is
possible to compare each mean with each other
mean using t-tests.
 However, conducting multiple t-tests can lead to
severe inflation of the Type I error rate.
 ANOVA can be used to test differences among
several means for significance without increasing
the Type I error rate.

7.  The ANOVA technique is important in the
context of all those situations where we want to
compare more than two populations.
 In such circumstances we do not want to consider
all possible combinations of two populations at a
time for that would require a great number of
tests before we would be able to arrive at a
decision.

8.  ANOVA is a procedure for testing the difference
among different groups of data for homogeneity.
 “The essence of ANOVA is that the total amount
of variation in a set of data is broken down into
two types, that amount which can be attributed to
chance and that amount which can be attributed
to specified causes.”

9.  The basic principle of ANOVA is to test for
differences among the means of the populations
by examining the amount of variation within
each of these samples, relative to the amount of
variation between the samples.

10.  The experimental errors of data are normally
distributed.
 Equal variances between treatments i.e.
Homogeneity of variances Homoscedasticity.
 Independence of samples i.e. Each sample is
randomly selected and independent.

11.  One-way (or single factor) ANOVA:
 t is the simplest type of ANOVA, in which only
one source of variation, or factor, is investigated.
 It is an extension to three or more samples of the
t test procedure for use with two independent
samples.
 In another way t test for use with two
independent samples is a special case of one-way
analysis of variance.

12. The technique involves the following steps:
i. Obtain the mean of each sample i.e.,
ii. Find the mean of the sample means:
iii. Calculate the sum of squares for variance
between the samples (or SS between):
KXXXX ,...,,, 321
)(.
...321
KofSamplesNo
XXXX
X K

     22
22
2
11 ...betweenSS XXnXXnXXn KK 

13. iv. Calculate Mean Square (MS) between:
v. Calculate the sum of squares for variance
within the samples (or SS within):
vi. Calculate Mean Square (MS) within:
1)-(K
betweenSS
BetweenMS 
       
2
Ki
2
22i
2
11i X...XXwithinSS KXXX
k)-(n
withinSS
withinMS 

14. vii. Calculate SS for total variance:
 SS for total variance = SS between + SS within.
 The degrees of freedom for between and within
must add up to the degrees of freedom for total
variance i.e.,
(n – 1) = (k – 1) + (n – k)
  
2
ijXvariancefor total XSS

15. viii. Finally, F-ratio may be worked out as under:
 This ratio is used to judge whether the difference
among several sample means is significant or is
just a matter of sampling fluctuations.
within
between
ratio-
MS
MS
F 

21.  Two-way ANOVA technique is used when the data are
classified on the basis of two factors. For example, the
agricultural output may be classified on the basis of
different varieties of seeds and also on the basis of
different varieties of fertilizers used.
 A statistical test used to determine the effect
of two nominal predictor variables on a continuous
outcome variable.
 two-way ANOVA test analyzes the effect of the
independent variables on the expected outcome along
with their relationship to the outcome itself.
 two-way design may have repeated measurements of
each factor or may not have repeated values.

22.  Types of two-way ANOVA
ANOVA technique in context of two-way design
when repeated values are not there.
ANOVA technique in context of two-way design
when repeated values are there.

23. i. ANOVA technique in context of two-way design when
repeated values are not there.
 it includes calculation of residual or error variation by
subtraction, once we have calculated the sum of squares
for total variance and for variance between varieties of
one treatment as also for variance between varieties of the
other treatment.
ii. ANOVA technique in context of two-way design when
repeated values are there.
 we can obtain a separate independent measure of inherent
or smallest variations.
 interaction variation: Interaction is the measure of inter
relationship among the two different classifications.

24.  Graphical method for studying interaction in
two-way design.
For graphs we shall select one of the factors to be
used as the x-axis.
Then we plot the averages for all the samples on the
graph and connect the averages for each variety of
other factor by a distinct line.
If the connecting lines do not cross over each other,
then the graph indicates that there is no interaction.
but if the lines do cross, they indicate definite
interaction or inter-relation between the two factors.

25. The graph indicates that there is a significant interaction
because the different connecting lines for groups of
people do cross over each other. We find that A and B are
affected very similarly, but C is affected differently.

26.  Analysis of covariance (ANCOVA) allows to
compare one variable in 2 or more groups taking
into account (or to correct for) variability of other
variables, called covariates.
 It tests whether there is a significant difference
between groups after controlling for variance
explained by a covariate.

27.  Why ANCOVA:
 The object of experimental design in general happens
to be to ensure that the results observed may be
attributed to the treatment variable and to no other
causal circumstances.
For instance, the researcher studying one independent
variable, X, may wish to control the influence of some
uncontrolled variable (sometimes called the covariate
or the concomitant variables), Z, which is known to be
correlated with the dependent variable, Y, then he
should use the technique of analysis of covariance for
a valid evaluation of the outcome of the experiment.
“In psychology and education primary interest in the
analysis of covariance rests in its use as a procedure
for the statistical control of an uncontrolled variable.”

28.  The influence of uncontrolled variable is usually
removed by simple linear regression method and
the residual sums of squares are used to provide
variance estimates which in turn are used to make
tests of significance.
 covariance analysis consists in subtracting from
each individual score ( ) that portion of it ´ that is
predictable from uncontrolled variable ( ) and then
computing the usual analysis of variance on the
resulting (Y – Y´)’s.
 The adjustment to the degree of freedom is made
because of the fact that estimation using regression
method required loss of degree of freedom.
iY iY
iY

29.  There is some sort of relationship between.
 The dependent variable and the uncontrolled
variable. We also assume that this form of
relationship is the same in the various treatment
groups.
 Other assumptions are:
(i) Various treatment groups are selected at random from the
population.
(ii) The groups are homogeneous in variability.
(iii) The regression is linear and is same from group to group.